I've reworked this and added on is any of it correct?Totally confused-learned this today and I don't get it-Please help
Write the standard form of 9x^2 + 4y^2-72x+16y+124=0
I think the first step is to take the constant to right side so I have:
9x^2 + 4y^2-72x+19(x^2-8x+16) + 4(y^2 + 4y + 4) = -124+ 144+16
9(x+4)^2 + 4(y^2 + y) = 36
(x+4)^2/4 + (y^2+y)/9 = 1
sure looks like an ellipse in standard form to me
but (y^2 + 4y + 4) is (y+2)^2
9 x^2 -72 x + ?
9 (x^2 - 8 x + 16)
9 (x-4)^2
so
9x^2 + 4y^2-72x+16y+124=0
9 x^2 -72 x + 4 y^2 + 16 y = -124
9(x^2-8x) + 4(y^2+4 y) = -124
9 (x^2-8x+16) + 4(y^2+4y) = -124+9*16
9(x-4)^2 +4(y^2+4y) = 20
9(x-4)^2 + 4(y^2+4y+4) = 20+16
9(x-4)^2+4(y+2)^2 = 36
so
(x-4)^2/4 +(y+2)^2/9 = 1
ellipse with center at (4,-2)
semi axes of length 2 and 3
Good effort! Let's go through the steps to confirm if your rework is correct:
1. Start with the equation: 9x^2 + 4y^2 - 72x + 16y + 124 = 0.
2. Rewrite the equation by grouping the x terms and the y terms separately:
9x^2 - 72x + 4y^2 + 16y + 124 = 0.
3. Move the constant term (124) to the right side of the equation:
9x^2 - 72x + 4y^2 + 16y = -124.
4. Complete the square for the x terms. Take half of the coefficient of x (-72/2 = -36) and square it (36^2 = 1296). Add this value inside the parentheses. Do the same for the y terms, taking half of the coefficient of y (16/2 = 8) and squaring it (8^2 = 64). Add this value inside the parentheses as well:
9(x^2 - 8x + 16) + 4(y^2 + 4y + 4) = -124 + 1296 + 64.
5. Simplify both sides of the equation:
9(x - 4)^2 + 4(y + 2)^2 = 1236.
Looking at your attempt, it seems that you correctly completed the square for the x terms and y terms separately. However, there are a few mistakes in your calculations. The constant term on the right side should be -124 + 1296 + 64, which equals 1236, not -124 + 144 + 16. Also, when writing the final equation in standard form, you need to divide both sides by the constant term (1236) to isolate the squared terms on the left side:
(x - 4)^2/137.33 + (y + 2)^2/309 = 1.
Therefore, the correct standard form of the equation is:
(x - 4)^2/137.33 + (y + 2)^2/309 = 1.
Keep practicing, and you'll improve your skills in no time!